Non‐autonomous forms and invariance

We generalize the Beurling–Deny–Ouhabaz criterion for parabolic evolution equations governed by forms to the non‐autonomous, non‐homogeneous and semilinear case. Let be Hilbert spaces such that V is continuously and densely embedded in H and let be the operator associated with a bounded H‐elliptic form for all . Suppose is closed and convex and the orthogonal projection onto . Given and , we investigate when the solution of the non‐autonomous evolutionary problem remains in and show that this is the case if for a.e. . Moreover, we examine necessity of this condition and apply this result to a semilinear problem.     

Variable Hardy–Lorentz spaces

Let be a measurable function on with . We introduce the variable Hardy–Lorentz space for via the radial grand maximal function. Under the assumption that satisfies the log‐Hölder condition, we establish a version of Fefferman–Stein vector‐valued inequality in variable Lorentz space by interpolation. We also construct atomic decompositions for , and develop a theory of real interpolation and formulate the dual space of the variable Hardy–Lorentz space with and . As a byproduct, we obtain a new John–Nirenberg theorem. Furthermore, we get equivalent characterizations of the variable Hardy–Lorentz space by means of the Lusin area function, the Littlewood–Paley g‐function and the Littlewood–Paley ‐function. Finally, we investigate the boundedness of singular operators on for and .     

There are eight‐element orthogonal exponentials on the spatial Sierpinski gasket

The self‐affine measure corresponding to an expanding matrix and the digit set in the space is supported on the spatial Sierpinski gasket, where are the standard basis of unit column vectors in and . In the case and , it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space is at most “4”, where the number 4 is the best upper bound. That is, all the four‐element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five‐element orthogonal exponentials in . In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert space to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.     

Average distances between points in graph‐directed self‐similar fractals

We study several distinct notions of average distances between points belonging to graph‐directed self‐similar subsets of . In particular, we compute the average distance with respect to graph‐directed self‐similar measures, and with respect to the normalised Hausdorff measure. As an application of our main results, we compute the average distance between two points belonging to the Drobot–Turner set with respect to the normalised Hausdorff measure, i.e. we compute where s denotes the Hausdorff dimension of and is the s‐dimensional Hausdorff measure; here the Drobot–Turner set (introduced by Drobot & Turner in 1989) is defined as follows, namely, for positive integers N and m and a positive real number c, the Drobot–Turner set is the set of those real numbers for which any m consecutive base N digits in the N‐ary expansion of x sum up to at least c. For example, if , and , then our results show that where is the unique positive real number such that .     

Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as of all weak (energy) solutions of a class of equations with the following model representative: with prescribed global energy function Here , , , Ω is a bounded smooth domain, . Particularly, in the case it is proved that the solution u remains uniformly bounded as in an arbitrary subdomain and the sharp upper estimate of when has been obtained depending on and . In the case for all , sharp sufficient conditions on degeneration of near that guarantee the above mentioned boundedness for an arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of the solution when has been obtained.     

Existence and regularity of positive solutions of a degenerate elliptic problem

Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form in Ω, on , where Ω is a bounded smooth domain in , , are obtained via new embeddings of some weighted Sobolev spaces with singular weights and . It is seen that and admit many singular points in Ω. The main embedding results in this paper provide some generalizations of the well‐known Caffarelli–Kohn–Nirenberg inequality.     

On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

We construct a bounded C¹ domain Ω in for which the regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in such that the solution of in Ω and either on or on is contained in but not in for any . An analogous result holds for Sobolev spaces with .     

The Tarski irredundant basis theorem and the finite soluble groups

Denote by and , respectively, the smallest and the largest cardinality of a minimal generating set of a finite group G. The Tarski irredundant basis theorem implies that for every k with there exist a minimal generating set , an index and in G such that is again a minimal generating set of G. In this case we say that is an immediate descendant of ω. There are several examples of minimal generating sets of cardinality smaller than which have no immediate descendant and so it appears an interesting problem to investigate under which conditions an immediate descendant exists. In this paper we discuss this problem in the case of finite soluble groups.     

Noncommutative topology and Jordan operator algebras

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a “full” noncommutative topology in the ‐algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of ‐algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about for a left ideal L in a ‐algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in ‐algebras and various subspaces of ‐algebras, related to open and closed projections and technical ‐algebraic results of Brown.     

Recurrence of singularities for second order isotropic pseudodifferential operators

Let H be a self‐adjoint isotropic elliptic pseudodifferential operator of order 2. Denote by the solution of the Schrödinger equation with initial data . If u₀ is compactly supported the solution is smooth for small , but not for all t. We determine the wavefront set of in terms of the wavefront set of u₀ and the principal and subprincipal symbol of H.     

Operators from the Hardy space to the α‐Bloch space

Let be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*‐triple X. In this paper, we characterize the bounded weighted composition operators from the Hardy space into the α‐Bloch space on . Also, we show the multiplication operator from into is bounded. Finally, we show that there exist no isometric composition operators.     

Decomposition of integral self‐affine multi‐tiles

In contrast to the situation with self‐affine tiles, the representation of self‐affine multi‐tiles may not be unique (for a fixed dilation matrix). Let be an integral self‐affine multi‐tile associated with an integral, expansive matrix B and let K tile by translates of . In this work, we propose a stepwise method to decompose K into measure disjoint pieces  satisfying in such a way that the collection of sets forms an integral self‐affine collection associated with the matrix B and this with a minimum number of pieces . When used on a given measurable subset K which tiles by translates of , this decomposition terminates after finitely many steps if and only if the set K is an integral self‐affine multi‐tile. Furthermore, we show that the minimal decomposition we provide is unique.     

A Szegő limit theorem for translation‐invariant operators on polygons

We prove Szeg?‐type trace asymptotics for translation‐invariant operators on polygons. More precisely, consider a Fourier multiplier on with a sufficiently decaying, smooth symbol . Let be the interior of a polygon and, for , define its scaled version . Then we study the spectral asymptotics for the operator , the spatial restriction of A onto : for entire functions h with we provide a complete asymptotic expansion of as . These trace asymptotics consist of three terms that reflect the geometry of the polygon. If P is replaced by a domain with smooth boundary, a complete asymptotic expansion of the trace has been known for more than 30 years. However, for polygons the formula for the constant order term in the asymptotics is new. In particular, we show that each corner of the polygon produces an extra contribution; as a consequence, the constant order term exhibits an anomaly similar to the heat trace asymptotics for the Dirichlet Laplacian.     

On quasi‐infinitely divisible distributions with a point mass

An infinitely divisible distribution on is a probability measure μ such that the characteristic function has a Lévy–Khintchine representation with characteristic triplet , where ν is a Lévy measure, and . A natural extension of such distributions are quasi‐infinitely distributions. Instead of a Lévy measure, we assume that ν is a “signed Lévy measure”, for further information on the definition see [10]. We show that a distribution with and , where is the absolutely continuous part, is quasi‐infinitely divisible if and only if for every . We apply this to show that certain variance mixtures of mean zero normal distributions are quasi‐infinitely divisible distributions, and we give an example of a quasi‐infinitely divisible distribution that is not continuous but has infinite quasi‐Lévy measure. Furthermore, it is shown that replacing the signed Lévy measure by a seemingly more general complex Lévy measure does not lead to new distributions. Last but not least it is proven that the class of quasi‐infinitely divisible distributions is not open, but path‐connected in the space of probability measures with the Prokhorov metric.     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

Expected intrinsic volumes and facet numbers of random beta‐polytopes

Let be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: and We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of . Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when , respectively , we can also cover the models in which are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of and . One of the main tools used in the proofs is the Blaschke–Petkantschin formula.     

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

On heteroclinic solutions for BVPs involving ‐Laplacian operators without asymptotic or growth assumptions

In this paper we consider the second order discontinuous equation in the real line, with ? an increasing homeomorphism such that and , with , for , a L¹‐Carathéodory function and verifying an adequate relation. We remark that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities ? and f. Moreover, as far as we know, our main result is even new when , that is, for the equation      

Twisted Orlicz algebras,II

Let G be a locally compact group, let be a 2‐cocycle, and let () be a complementary pair of strictly increasing continuous Young functions. We continue our investigation in [14] of the algebraic properties of the Orlicz space with respect to the twisted convolution ? coming from Ω. We show that the twisted Orlicz algebra posses a bounded approximate identity if and only if it is unital if and only if G is discrete. On the other hand, under suitable condition on Ω, becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of , namely amenability and Connes‐amenability, and show that they rarely happen. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.